Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality

Abstract: In contrast to finite dimensions where disordered systems display multifractal statistics only at criticality, the tree geometry induces multifractal statistics for disordered systems also off criticality. For the Anderson tight-binding localization model defined on a tree of branching ratio K=2 with $N$ generations, we consider the Miller-Derrida scattering geometry [J. Stat. Phys. 75, 357 (1994)], where an incoming wire is attached to the root of the tree, and where $K^{N}$ outcoming wires are attached to the leaves of the tree. In terms of the $K^{N}$ transmission amplitudes $t_j$, the total Landauer transmission is $T \equiv \sum_j | t_j |^2$, so that each channel $j$ is characterized by the weight $w_j=| t_j |^2/T$. We numerically measure the typical multifractal singularity spectrum $f(\alpha)$ of these weights as a function of the disorder strength $W$ and we obtain the following conclusions for its left-termination point $\alpha_+(W)$. In the delocalized phase $W<W_c$, $\alpha_+(W)$ is strictly positive $\alpha_+(W)\>0$ and is associated with a moment index $q_+(W)>1$. At criticality, it vanishes $\alpha_+(W_c)=0$ and is associated with the moment index $q_+(W_c)=1$. In the localized phase $W>W_c$, $\alpha_+(W)=0$ is associated with some moment index $q_+(W)<1$. We discuss the similarities with the exact results concerning the multifractal properties of the Directed Polymer on the Cayley tree.